A349519 -id:A349519 - cp's OEIS frontend

A349518 a(n)=x is the least prime with pi(x,4,3) - pi(x,4,1) = n-1 where pi(x,4,k) is the number of primes 4*j + k <= x.

2, 3, 11, 71, 83, 223, 227, 503, 751, 1531, 1543, 1571, 1579, 1583, 4127, 5147, 5171, 5179, 5651, 6211, 11083, 11087, 11471, 11483, 11519, 11527, 11579, 11587, 17239, 17551, 17903, 17971, 35963, 36011, 39703, 39727

Offset: 1

Gerhard Kirchner, Nov 20 2021

nonn

The difference d(x) = pi(x,4,3) - pi(x,4,1) changes sign infinitely often, see link "Prime Quadratic Effect". But this does not say anything about the amplitudes of these oscillations. For diagrams, see link "Oscillations of d(x)". If d(x) has no upper limit, the current sequence is infinite. Regarding the lower limit, see A349519.

			primes 4*j+1: 5, 13, 17, ...
       4*j+3: 3, 7, 11, ...
d(x) = pi(x,4,3) - pi(x,4,1)
.
  n  x  pi(x,4,3) pi(x,4,1)  d(x)=n-1?
  - --  --------- ---------  ---------
  1  2       0         0     0=0  true   a(1)  = 2
  2  3       1         0     1=1  true   a(2)  = 3
  3  5       1         1     0=2 false   a(3) != 5
  ...........................
  3 11       3         1     2=2  true   a(3)  = 11

Gerhard Kirchner, Oscillations of d(x)
Eric Weisstein's World of Mathematics, Prime Quadratic Effect.

Cf. A038691, A007350.

block(w:[2],  su:0, sum:0, n:1, p:2, nmax: 40,
  /* returns nmax terms */
while nsum then(n:n+1, sum:su, w: append(w,[p]) ) ) ,
w);

cp's OEIS Frontend

A349518 a(n)=x is the least prime with pi(x,4,3) - pi(x,4,1) = n-1 where pi(x,4,k) is the number of primes 4*j + k <= x.

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